LMFDB - Embedded newform 9405.2.a.v.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [9405,2,Mod(1,9405)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("9405.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9405, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 9405 = 3^{2} \cdot 5 \cdot 11 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9405.a (trivial)

Newform invariants

comment:select newform

sage:traces = [5,3,0,5,-5,0,-11,-3,0,-3,5] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$75.0993031010$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	$\Q(\zeta_{22})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 4x^{3} + 3x^{2} + 3x - 1 $ x^5 - x^4 - 4x^3 + 3x^2 + 3*x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1045)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-0.830830$ of defining polynomial
Character	$\chi$	$=$	9405.1

$q$-expansion

comment:q-expansion

sage:f.q_expansion() # note that sage often uses an isomorphic number field

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.37279 q^{2} -0.115460 q^{4} -1.00000 q^{5} +2.43232 q^{7} -2.90407 q^{8} -1.37279 q^{10} +1.00000 q^{11} -4.84815 q^{13} +3.33906 q^{14} -3.75575 q^{16} +1.04001 q^{17} +1.00000 q^{19} +0.115460 q^{20} +1.37279 q^{22} -0.377030 q^{23} +1.00000 q^{25} -6.65547 q^{26} -0.280835 q^{28} -4.50688 q^{29} +4.76341 q^{31} +0.652306 q^{32} +1.42770 q^{34} -2.43232 q^{35} +3.01428 q^{37} +1.37279 q^{38} +2.90407 q^{40} -0.790451 q^{41} -7.46777 q^{43} -0.115460 q^{44} -0.517582 q^{46} +9.67305 q^{47} -1.08380 q^{49} +1.37279 q^{50} +0.559766 q^{52} +12.7465 q^{53} -1.00000 q^{55} -7.06364 q^{56} -6.18698 q^{58} -1.32111 q^{59} -5.58779 q^{61} +6.53915 q^{62} +8.40698 q^{64} +4.84815 q^{65} -1.20728 q^{67} -0.120079 q^{68} -3.33906 q^{70} +0.259538 q^{71} -6.27686 q^{73} +4.13797 q^{74} -0.115460 q^{76} +2.43232 q^{77} -9.16926 q^{79} +3.75575 q^{80} -1.08512 q^{82} +14.2430 q^{83} -1.04001 q^{85} -10.2517 q^{86} -2.90407 q^{88} +1.23983 q^{89} -11.7923 q^{91} +0.0435318 q^{92} +13.2790 q^{94} -1.00000 q^{95} +2.95633 q^{97} -1.48783 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 3 q^{2} + 5 q^{4} - 5 q^{5} - 11 q^{7} - 3 q^{8} - 3 q^{10} + 5 q^{11} + q^{13} - 3 q^{16} + 3 q^{17} + 5 q^{19} - 5 q^{20} + 3 q^{22} + 8 q^{23} + 5 q^{25} + 16 q^{26} - 22 q^{28} - 11 q^{29} - 5 q^{31}+ \cdots - 10 q^{98}+O(q^{100}) $ 5 * q + 3 * q^2 + 5 * q^4 - 5 * q^5 - 11 * q^7 - 3 * q^8 - 3 * q^10 + 5 * q^11 + q^13 - 3 * q^16 + 3 * q^17 + 5 * q^19 - 5 * q^20 + 3 * q^22 + 8 * q^23 + 5 * q^25 + 16 * q^26 - 22 * q^28 - 11 * q^29 - 5 * q^31 + 2 * q^32 + 4 * q^34 + 11 * q^35 - 9 * q^37 + 3 * q^38 + 3 * q^40 - 15 * q^41 - 13 * q^43 + 5 * q^44 + 18 * q^46 + 20 * q^47 + 20 * q^49 + 3 * q^50 + q^52 + 5 * q^53 - 5 * q^55 - 33 * q^58 + 17 * q^59 + 3 * q^61 - 14 * q^62 - 17 * q^64 - q^65 - 28 * q^67 + 25 * q^68 + 6 * q^71 - 16 * q^73 + 21 * q^74 + 5 * q^76 - 11 * q^77 + 3 * q^79 + 3 * q^80 + 2 * q^82 + 33 * q^83 - 3 * q^85 - 10 * q^86 - 3 * q^88 + 16 * q^89 - 22 * q^91 + 19 * q^92 - 10 * q^94 - 5 * q^95 - 14 * q^97 - 10 * q^98

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	1.37279	0.970706	0.485353	−	0.874318i	$-0.338691\pi$
$2$	1.37279	0.970706	0.485353	+	0.874318i	$0.338691\pi$
$3$	0	0
$3$	0	0
$4$	−0.115460	−0.0577299
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	2.43232	0.919332	0.459666	−	0.888092i	$-0.347969\pi$
$7$	2.43232	0.919332	0.459666	+	0.888092i	$0.347969\pi$
$8$	−2.90407	−1.02674
$9$	0	0
$10$	−1.37279	−0.434113
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	−4.84815	−1.34463	−0.672317	−	0.740263i	$-0.734701\pi$
$13$	−4.84815	−1.34463	−0.672317	+	0.740263i	$0.734701\pi$
$14$	3.33906	0.892401
$15$	0	0
$16$	−3.75575	−0.938937
$17$	1.04001	0.252238	0.126119	−	0.992015i	$-0.459748\pi$
$17$	1.04001	0.252238	0.126119	+	0.992015i	$0.459748\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0.115460	0.0258176
$21$	0	0
$22$	1.37279	0.292679
$23$	−0.377030	−0.0786163	−0.0393081	−	0.999227i	$-0.512515\pi$
$23$	−0.377030	−0.0786163	−0.0393081	+	0.999227i	$0.512515\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−6.65547	−1.30525
$27$	0	0
$28$	−0.280835	−0.0530729
$29$	−4.50688	−0.836907	−0.418453	−	0.908238i	$-0.637428\pi$
$29$	−4.50688	−0.836907	−0.418453	+	0.908238i	$0.637428\pi$
$30$	0	0
$31$	4.76341	0.855535	0.427767	−	0.903889i	$-0.359300\pi$
$31$	4.76341	0.855535	0.427767	+	0.903889i	$0.359300\pi$
$32$	0.652306	0.115313
$33$	0	0
$34$	1.42770	0.244849
$35$	−2.43232	−0.411138
$36$	0	0
$37$	3.01428	0.495545	0.247773	−	0.968818i	$-0.420301\pi$
$37$	3.01428	0.495545	0.247773	+	0.968818i	$0.420301\pi$
$38$	1.37279	0.222695
$39$	0	0
$40$	2.90407	0.459174
$41$	−0.790451	−0.123448	−0.0617238	−	0.998093i	$-0.519660\pi$
$41$	−0.790451	−0.123448	−0.0617238	+	0.998093i	$0.519660\pi$
$42$	0	0
$43$	−7.46777	−1.13882	−0.569412	−	0.822052i	$-0.692829\pi$
$43$	−7.46777	−1.13882	−0.569412	+	0.822052i	$0.692829\pi$
$44$	−0.115460	−0.0174062
$45$	0	0
$46$	−0.517582	−0.0763133
$47$	9.67305	1.41096	0.705479	−	0.708730i	$-0.250732\pi$
$47$	9.67305	1.41096	0.705479	+	0.708730i	$0.250732\pi$
$48$	0	0
$49$	−1.08380	−0.154829
$50$	1.37279	0.194141
$51$	0	0
$52$	0.559766	0.0776256
$53$	12.7465	1.75087	0.875435	−	0.483336i	$-0.160575\pi$
$53$	12.7465	1.75087	0.875435	+	0.483336i	$0.160575\pi$
$54$	0	0
$55$	−1.00000	−0.134840
$56$	−7.06364	−0.943919
$57$	0	0
$58$	−6.18698	−0.812390
$59$	−1.32111	−0.171994	−0.0859968	−	0.996295i	$-0.527407\pi$
$59$	−1.32111	−0.171994	−0.0859968	+	0.996295i	$0.527407\pi$
$60$	0	0
$61$	−5.58779	−0.715443	−0.357721	−	0.933828i	$-0.616446\pi$
$61$	−5.58779	−0.715443	−0.357721	+	0.933828i	$0.616446\pi$
$62$	6.53915	0.830472
$63$	0	0
$64$	8.40698	1.05087
$65$	4.84815	0.601339
$66$	0	0
$67$	−1.20728	−0.147493	−0.0737465	−	0.997277i	$-0.523496\pi$
$67$	−1.20728	−0.147493	−0.0737465	+	0.997277i	$0.523496\pi$
$68$	−0.120079	−0.0145617
$69$	0	0
$70$	−3.33906	−0.399094
$71$	0.259538	0.0308015	0.0154007	−	0.999881i	$-0.495098\pi$
$71$	0.259538	0.0308015	0.0154007	+	0.999881i	$0.495098\pi$
$72$	0	0
$73$	−6.27686	−0.734651	−0.367325	−	0.930093i	$-0.619726\pi$
$73$	−6.27686	−0.734651	−0.367325	+	0.930093i	$0.619726\pi$
$74$	4.13797	0.481029
$75$	0	0
$76$	−0.115460	−0.0132441
$77$	2.43232	0.277189
$78$	0	0
$79$	−9.16926	−1.03162	−0.515811	−	0.856702i	$-0.672509\pi$
$79$	−9.16926	−1.03162	−0.515811	+	0.856702i	$0.672509\pi$
$80$	3.75575	0.419906
$81$	0	0
$82$	−1.08512	−0.119831
$83$	14.2430	1.56337	0.781686	−	0.623673i	$-0.214360\pi$
$83$	14.2430	1.56337	0.781686	+	0.623673i	$0.214360\pi$
$84$	0	0
$85$	−1.04001	−0.112804
$86$	−10.2517	−1.10546
$87$	0	0
$88$	−2.90407	−0.309575
$89$	1.23983	0.131421	0.0657107	−	0.997839i	$-0.479069\pi$
$89$	1.23983	0.131421	0.0657107	+	0.997839i	$0.479069\pi$
$90$	0	0
$91$	−11.7923	−1.23617
$92$	0.0435318	0.00453851
$93$	0	0
$94$	13.2790	1.36963
$95$	−1.00000	−0.102598
$96$	0	0
$97$	2.95633	0.300170	0.150085	−	0.988673i	$-0.452045\pi$
$97$	2.95633	0.300170	0.150085	+	0.988673i	$0.452045\pi$
$98$	−1.48783	−0.150294
$99$	0	0
$100$	−0.115460	−0.0115460
$101$	1.55010	0.154241	0.0771204	−	0.997022i	$-0.475427\pi$
$101$	1.55010	0.154241	0.0771204	+	0.997022i	$0.475427\pi$
$102$	0	0
$103$	−0.339313	−0.0334335	−0.0167168	−	0.999860i	$-0.505321\pi$
$103$	−0.339313	−0.0334335	−0.0167168	+	0.999860i	$0.505321\pi$
$104$	14.0794	1.38060
$105$	0	0
$106$	17.4982	1.69958
$107$	9.05124	0.875017	0.437508	−	0.899214i	$-0.355861\pi$
$107$	9.05124	0.875017	0.437508	+	0.899214i	$0.355861\pi$
$108$	0	0
$109$	14.7515	1.41294	0.706471	−	0.707742i	$-0.250286\pi$
$109$	14.7515	1.41294	0.706471	+	0.707742i	$0.250286\pi$
$110$	−1.37279	−0.130890
$111$	0	0
$112$	−9.13520	−0.863195
$113$	4.71030	0.443108	0.221554	−	0.975148i	$-0.428887\pi$
$113$	4.71030	0.443108	0.221554	+	0.975148i	$0.428887\pi$
$114$	0	0
$115$	0.377030	0.0351583
$116$	0.520363	0.0483145
$117$	0	0
$118$	−1.81360	−0.166955
$119$	2.52963	0.231891
$120$	0	0
$121$	1.00000	0.0909091
$122$	−7.67083	−0.694485
$123$	0	0
$124$	−0.549982	−0.0493899
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	14.7860	1.31205	0.656024	−	0.754740i	$-0.272237\pi$
$127$	14.7860	1.31205	0.656024	+	0.754740i	$0.272237\pi$
$128$	10.2364	0.904775
$129$	0	0
$130$	6.65547	0.583723
$131$	7.82065	0.683294	0.341647	−	0.939828i	$-0.389015\pi$
$131$	7.82065	0.683294	0.341647	+	0.939828i	$0.389015\pi$
$132$	0	0
$133$	2.43232	0.210909
$134$	−1.65734	−0.143172
$135$	0	0
$136$	−3.02025	−0.258984
$137$	7.70625	0.658389	0.329195	−	0.944262i	$-0.393223\pi$
$137$	7.70625	0.658389	0.329195	+	0.944262i	$0.393223\pi$
$138$	0	0
$139$	14.3953	1.22099	0.610495	−	0.792020i	$-0.290970\pi$
$139$	14.3953	1.22099	0.610495	+	0.792020i	$0.290970\pi$
$140$	0.280835	0.0237349
$141$	0	0
$142$	0.356290	0.0298992
$143$	−4.84815	−0.405423
$144$	0	0
$145$	4.50688	0.374276
$146$	−8.61678	−0.713130
$147$	0	0
$148$	−0.348028	−0.0286078
$149$	13.8399	1.13381	0.566906	−	0.823783i	$-0.308140\pi$
$149$	13.8399	1.13381	0.566906	+	0.823783i	$0.308140\pi$
$150$	0	0
$151$	−8.10281	−0.659398	−0.329699	−	0.944086i	$-0.606947\pi$
$151$	−8.10281	−0.659398	−0.329699	+	0.944086i	$0.606947\pi$
$152$	−2.90407	−0.235551
$153$	0	0
$154$	3.33906	0.269069
$155$	−4.76341	−0.382607
$156$	0	0
$157$	−7.13378	−0.569338	−0.284669	−	0.958626i	$-0.591884\pi$
$157$	−7.13378	−0.569338	−0.284669	+	0.958626i	$0.591884\pi$
$158$	−12.5874	−1.00140
$159$	0	0
$160$	−0.652306	−0.0515693
$161$	−0.917060	−0.0722744
$162$	0	0
$163$	−23.8705	−1.86968	−0.934840	−	0.355068i	$-0.884458\pi$
$163$	−23.8705	−1.86968	−0.934840	+	0.355068i	$0.884458\pi$
$164$	0.0912652	0.00712661
$165$	0	0
$166$	19.5526	1.51757
$167$	8.49981	0.657735	0.328868	−	0.944376i	$-0.393333\pi$
$167$	8.49981	0.657735	0.328868	+	0.944376i	$0.393333\pi$
$168$	0	0
$169$	10.5046	0.808043
$170$	−1.42770	−0.109500
$171$	0	0
$172$	0.862227	0.0657442
$173$	3.76732	0.286424	0.143212	−	0.989692i	$-0.454257\pi$
$173$	3.76732	0.286424	0.143212	+	0.989692i	$0.454257\pi$
$174$	0	0
$175$	2.43232	0.183866
$176$	−3.75575	−0.283100
$177$	0	0
$178$	1.70202	0.127571
$179$	12.0321	0.899323	0.449661	−	0.893199i	$-0.351545\pi$
$179$	12.0321	0.899323	0.449661	+	0.893199i	$0.351545\pi$
$180$	0	0
$181$	−9.01903	−0.670380	−0.335190	−	0.942151i	$-0.608800\pi$
$181$	−9.01903	−0.670380	−0.335190	+	0.942151i	$0.608800\pi$
$182$	−16.1883	−1.19995
$183$	0	0
$184$	1.09492	0.0807188
$185$	−3.01428	−0.221615
$186$	0	0
$187$	1.04001	0.0760527
$188$	−1.11685	−0.0814544
$189$	0	0
$190$	−1.37279	−0.0995923
$191$	15.2598	1.10416	0.552079	−	0.833792i	$-0.313835\pi$
$191$	15.2598	1.10416	0.552079	+	0.833792i	$0.313835\pi$
$192$	0	0
$193$	3.17246	0.228359	0.114179	−	0.993460i	$-0.463576\pi$
$193$	3.17246	0.228359	0.114179	+	0.993460i	$0.463576\pi$
$194$	4.05841	0.291377
$195$	0	0
$196$	0.125136	0.00893827
$197$	6.09397	0.434178	0.217089	−	0.976152i	$-0.430344\pi$
$197$	6.09397	0.434178	0.217089	+	0.976152i	$0.430344\pi$
$198$	0	0
$199$	−15.8148	−1.12108	−0.560541	−	0.828127i	$-0.689407\pi$
$199$	−15.8148	−1.12108	−0.560541	+	0.828127i	$0.689407\pi$
$200$	−2.90407	−0.205349
$201$	0	0
$202$	2.12796	0.149723
$203$	−10.9622	−0.769395
$204$	0	0
$205$	0.790451	0.0552075
$206$	−0.465805	−0.0324541
$207$	0	0
$208$	18.2084	1.26253
$209$	1.00000	0.0691714
$210$	0	0
$211$	10.4371	0.718522	0.359261	−	0.933237i	$-0.383029\pi$
$211$	10.4371	0.718522	0.359261	+	0.933237i	$0.383029\pi$
$212$	−1.47171	−0.101077
$213$	0	0
$214$	12.4254	0.849384
$215$	7.46777	0.509298
$216$	0	0
$217$	11.5862	0.786520
$218$	20.2507	1.37155
$219$	0	0
$220$	0.115460	0.00778429
$221$	−5.04210	−0.339168
$222$	0	0
$223$	−15.2161	−1.01895	−0.509474	−	0.860486i	$-0.670160\pi$
$223$	−15.2161	−1.01895	−0.509474	+	0.860486i	$0.670160\pi$
$224$	1.58662	0.106010
$225$	0	0
$226$	6.46623	0.430127
$227$	15.4170	1.02326	0.511630	−	0.859206i	$-0.329042\pi$
$227$	15.4170	1.02326	0.511630	+	0.859206i	$0.329042\pi$
$228$	0	0
$229$	18.2529	1.20618	0.603091	−	0.797672i	$-0.293935\pi$
$229$	18.2529	1.20618	0.603091	+	0.797672i	$0.293935\pi$
$230$	0.517582	0.0341283
$231$	0	0
$232$	13.0883	0.859289
$233$	−19.1041	−1.25155	−0.625776	−	0.780003i	$-0.715218\pi$
$233$	−19.1041	−1.25155	−0.625776	+	0.780003i	$0.715218\pi$
$234$	0	0
$235$	−9.67305	−0.631000
$236$	0.152535	0.00992917
$237$	0	0
$238$	3.47264	0.225098
$239$	−20.6551	−1.33607	−0.668034	−	0.744131i	$-0.732864\pi$
$239$	−20.6551	−1.33607	−0.668034	+	0.744131i	$0.732864\pi$
$240$	0	0
$241$	15.9963	1.03042	0.515208	−	0.857065i	$-0.327715\pi$
$241$	15.9963	1.03042	0.515208	+	0.857065i	$0.327715\pi$
$242$	1.37279	0.0882460
$243$	0	0
$244$	0.645164	0.0413024
$245$	1.08380	0.0692417
$246$	0	0
$247$	−4.84815	−0.308480
$248$	−13.8333	−0.878416
$249$	0	0
$250$	−1.37279	−0.0868226
$251$	11.3379	0.715639	0.357820	−	0.933791i	$-0.383520\pi$
$251$	11.3379	0.715639	0.357820	+	0.933791i	$0.383520\pi$
$252$	0	0
$253$	−0.377030	−0.0237037
$254$	20.2981	1.27361
$255$	0	0
$256$	−2.76162	−0.172601
$257$	−9.80023	−0.611322	−0.305661	−	0.952140i	$-0.598877\pi$
$257$	−9.80023	−0.611322	−0.305661	+	0.952140i	$0.598877\pi$
$258$	0	0
$259$	7.33171	0.455570
$260$	−0.559766	−0.0347152
$261$	0	0
$262$	10.7361	0.663277
$263$	21.2899	1.31279	0.656395	−	0.754418i	$-0.272081\pi$
$263$	21.2899	1.31279	0.656395	+	0.754418i	$0.272081\pi$
$264$	0	0
$265$	−12.7465	−0.783013
$266$	3.33906	0.204731
$267$	0	0
$268$	0.139392	0.00851475
$269$	18.2788	1.11448	0.557240	−	0.830352i	$-0.311860\pi$
$269$	18.2788	1.11448	0.557240	+	0.830352i	$0.311860\pi$
$270$	0	0
$271$	19.7872	1.20199	0.600994	−	0.799253i	$-0.294772\pi$
$271$	19.7872	1.20199	0.600994	+	0.799253i	$0.294772\pi$
$272$	−3.90600	−0.236836
$273$	0	0
$274$	10.5790	0.639103
$275$	1.00000	0.0603023
$276$	0	0
$277$	24.5658	1.47602	0.738008	−	0.674792i	$-0.235767\pi$
$277$	24.5658	1.47602	0.738008	+	0.674792i	$0.235767\pi$
$278$	19.7616	1.18522
$279$	0	0
$280$	7.06364	0.422133
$281$	−14.0898	−0.840524	−0.420262	−	0.907403i	$-0.638062\pi$
$281$	−14.0898	−0.840524	−0.420262	+	0.907403i	$0.638062\pi$
$282$	0	0
$283$	−18.8367	−1.11973	−0.559863	−	0.828585i	$-0.689146\pi$
$283$	−18.8367	−1.11973	−0.559863	+	0.828585i	$0.689146\pi$
$284$	−0.0299662	−0.00177816
$285$	0	0
$286$	−6.65547	−0.393546
$287$	−1.92263	−0.113489
$288$	0	0
$289$	−15.9184	−0.936376
$290$	6.18698	0.363312
$291$	0	0
$292$	0.724724	0.0424113
$293$	−2.08270	−0.121673	−0.0608364	−	0.998148i	$-0.519377\pi$
$293$	−2.08270	−0.121673	−0.0608364	+	0.998148i	$0.519377\pi$
$294$	0	0
$295$	1.32111	0.0769179
$296$	−8.75370	−0.508798
$297$	0	0
$298$	18.9993	1.10060
$299$	1.82790	0.105710
$300$	0	0
$301$	−18.1640	−1.04696
$302$	−11.1234	−0.640081
$303$	0	0
$304$	−3.75575	−0.215407
$305$	5.58779	0.319956
$306$	0	0
$307$	19.8442	1.13257	0.566284	−	0.824210i	$-0.308381\pi$
$307$	19.8442	1.13257	0.566284	+	0.824210i	$0.308381\pi$
$308$	−0.280835	−0.0160021
$309$	0	0
$310$	−6.53915	−0.371399
$311$	−3.71814	−0.210836	−0.105418	−	0.994428i	$-0.533618\pi$
$311$	−3.71814	−0.210836	−0.105418	+	0.994428i	$0.533618\pi$
$312$	0	0
$313$	27.0866	1.53102	0.765511	−	0.643423i	$-0.222486\pi$
$313$	27.0866	1.53102	0.765511	+	0.643423i	$0.222486\pi$
$314$	−9.79316	−0.552660
$315$	0	0
$316$	1.05868	0.0595554
$317$	−9.50445	−0.533823	−0.266911	−	0.963721i	$-0.586003\pi$
$317$	−9.50445	−0.533823	−0.266911	+	0.963721i	$0.586003\pi$
$318$	0	0
$319$	−4.50688	−0.252337
$320$	−8.40698	−0.469964
$321$	0	0
$322$	−1.25893	−0.0701572
$323$	1.04001	0.0578674
$324$	0	0
$325$	−4.84815	−0.268927
$326$	−32.7691	−1.81491
$327$	0	0
$328$	2.29553	0.126749
$329$	23.5280	1.29714
$330$	0	0
$331$	7.27869	0.400073	0.200036	−	0.979788i	$-0.435894\pi$
$331$	7.27869	0.400073	0.200036	+	0.979788i	$0.435894\pi$
$332$	−1.64449	−0.0902532
$333$	0	0
$334$	11.6684	0.638467
$335$	1.20728	0.0659609
$336$	0	0
$337$	29.3790	1.60037	0.800187	−	0.599750i	$-0.204733\pi$
$337$	29.3790	1.60037	0.800187	+	0.599750i	$0.204733\pi$
$338$	14.4205	0.784372
$339$	0	0
$340$	0.120079	0.00651218
$341$	4.76341	0.257953
$342$	0	0
$343$	−19.6624	−1.06167
$344$	21.6870	1.16928
$345$	0	0
$346$	5.17173	0.278034
$347$	−28.1028	−1.50864	−0.754318	−	0.656509i	$-0.772032\pi$
$347$	−28.1028	−1.50864	−0.754318	+	0.656509i	$0.772032\pi$
$348$	0	0
$349$	20.9948	1.12383	0.561913	−	0.827196i	$-0.310065\pi$
$349$	20.9948	1.12383	0.561913	+	0.827196i	$0.310065\pi$
$350$	3.33906	0.178480
$351$	0	0
$352$	0.652306	0.0347680
$353$	−15.1874	−0.808345	−0.404172	−	0.914683i	$-0.632440\pi$
$353$	−15.1874	−0.808345	−0.404172	+	0.914683i	$0.632440\pi$
$354$	0	0
$355$	−0.259538	−0.0137748
$356$	−0.143150	−0.00758693
$357$	0	0
$358$	16.5175	0.872978
$359$	−11.5028	−0.607093	−0.303546	−	0.952817i	$-0.598171\pi$
$359$	−11.5028	−0.607093	−0.303546	+	0.952817i	$0.598171\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	−12.3812	−0.650741
$363$	0	0
$364$	1.36153	0.0713637
$365$	6.27686	0.328546
$366$	0	0
$367$	−1.52355	−0.0795284	−0.0397642	−	0.999209i	$-0.512661\pi$
$367$	−1.52355	−0.0795284	−0.0397642	+	0.999209i	$0.512661\pi$
$368$	1.41603	0.0738158
$369$	0	0
$370$	−4.13797	−0.215123
$371$	31.0037	1.60963
$372$	0	0
$373$	16.5646	0.857682	0.428841	−	0.903380i	$-0.358922\pi$
$373$	16.5646	0.857682	0.428841	+	0.903380i	$0.358922\pi$
$374$	1.42770	0.0738248
$375$	0	0
$376$	−28.0912	−1.44869
$377$	21.8500	1.12533
$378$	0	0
$379$	−0.0190602	−0.000979058 0	−0.000489529	−	1.00000i	$-0.500156\pi$
$379$	−0.0190602	−0.000979058 0	−0.000489529		1.00000i	$0.500156\pi$
$380$	0.115460	0.00592296
$381$	0	0
$382$	20.9484	1.07181
$383$	−18.2085	−0.930412	−0.465206	−	0.885202i	$-0.654020\pi$
$383$	−18.2085	−0.930412	−0.465206	+	0.885202i	$0.654020\pi$
$384$	0	0
$385$	−2.43232	−0.123963
$386$	4.35511	0.221669
$387$	0	0
$388$	−0.341337	−0.0173288
$389$	31.4031	1.59220	0.796099	−	0.605166i	$-0.206893\pi$
$389$	31.4031	1.59220	0.796099	+	0.605166i	$0.206893\pi$
$390$	0	0
$391$	−0.392114	−0.0198300
$392$	3.14745	0.158970
$393$	0	0
$394$	8.36572	0.421459
$395$	9.16926	0.461355
$396$	0	0
$397$	−25.8619	−1.29797	−0.648985	−	0.760801i	$-0.724806\pi$
$397$	−25.8619	−1.29797	−0.648985	+	0.760801i	$0.724806\pi$
$398$	−21.7103	−1.08824
$399$	0	0
$400$	−3.75575	−0.187787
$401$	15.5141	0.774737	0.387369	−	0.921925i	$-0.373384\pi$
$401$	15.5141	0.774737	0.387369	+	0.921925i	$0.373384\pi$
$402$	0	0
$403$	−23.0938	−1.15038
$404$	−0.178974	−0.00890430
$405$	0	0
$406$	−15.0487	−0.746856
$407$	3.01428	0.149412
$408$	0	0
$409$	20.7451	1.02578	0.512889	−	0.858455i	$-0.328575\pi$
$409$	20.7451	1.02578	0.512889	+	0.858455i	$0.328575\pi$
$410$	1.08512	0.0535902
$411$	0	0
$412$	0.0391770	0.00193011
$413$	−3.21336	−0.158119
$414$	0	0
$415$	−14.2430	−0.699161
$416$	−3.16248	−0.155053
$417$	0	0
$418$	1.37279	0.0671451
$419$	−6.94955	−0.339508	−0.169754	−	0.985487i	$-0.554297\pi$
$419$	−6.94955	−0.339508	−0.169754	+	0.985487i	$0.554297\pi$
$420$	0	0
$421$	17.0486	0.830897	0.415448	−	0.909617i	$-0.363625\pi$
$421$	17.0486	0.830897	0.415448	+	0.909617i	$0.363625\pi$
$422$	14.3280	0.697474
$423$	0	0
$424$	−37.0168	−1.79770
$425$	1.04001	0.0504477
$426$	0	0
$427$	−13.5913	−0.657729
$428$	−1.04505	−0.0505146
$429$	0	0
$430$	10.2517	0.494378
$431$	−10.6174	−0.511421	−0.255711	−	0.966753i	$-0.582309\pi$
$431$	−10.6174	−0.511421	−0.255711	+	0.966753i	$0.582309\pi$
$432$	0	0
$433$	−5.21952	−0.250834	−0.125417	−	0.992104i	$-0.540027\pi$
$433$	−5.21952	−0.250834	−0.125417	+	0.992104i	$0.540027\pi$
$434$	15.9053	0.763480
$435$	0	0
$436$	−1.70321	−0.0815689
$437$	−0.377030	−0.0180358
$438$	0	0
$439$	−25.8096	−1.23182	−0.615912	−	0.787815i	$-0.711212\pi$
$439$	−25.8096	−1.23182	−0.615912	+	0.787815i	$0.711212\pi$
$440$	2.90407	0.138446
$441$	0	0
$442$	−6.92172	−0.329233
$443$	30.3917	1.44395	0.721977	−	0.691917i	$-0.243234\pi$
$443$	30.3917	1.44395	0.721977	+	0.691917i	$0.243234\pi$
$444$	0	0
$445$	−1.23983	−0.0587734
$446$	−20.8885	−0.989099
$447$	0	0
$448$	20.4485	0.966100
$449$	−10.8574	−0.512394	−0.256197	−	0.966625i	$-0.582470\pi$
$449$	−10.8574	−0.512394	−0.256197	+	0.966625i	$0.582470\pi$
$450$	0	0
$451$	−0.790451	−0.0372209
$452$	−0.543850	−0.0255805
$453$	0	0
$454$	21.1642	0.993285
$455$	11.7923	0.552830
$456$	0	0
$457$	18.3849	0.860008	0.430004	−	0.902827i	$-0.358512\pi$
$457$	18.3849	0.860008	0.430004	+	0.902827i	$0.358512\pi$
$458$	25.0573	1.17085
$459$	0	0
$460$	−0.0435318	−0.00202968
$461$	28.2662	1.31649	0.658245	−	0.752804i	$-0.271299\pi$
$461$	28.2662	1.31649	0.658245	+	0.752804i	$0.271299\pi$
$462$	0	0
$463$	33.4266	1.55346	0.776732	−	0.629831i	$-0.216876\pi$
$463$	33.4266	1.55346	0.776732	+	0.629831i	$0.216876\pi$
$464$	16.9267	0.785803
$465$	0	0
$466$	−26.2258	−1.21489
$467$	1.21297	0.0561297	0.0280648	−	0.999606i	$-0.491066\pi$
$467$	1.21297	0.0561297	0.0280648	+	0.999606i	$0.491066\pi$
$468$	0	0
$469$	−2.93650	−0.135595
$470$	−13.2790	−0.612515
$471$	0	0
$472$	3.83659	0.176594
$473$	−7.46777	−0.343369
$474$	0	0
$475$	1.00000	0.0458831
$476$	−0.292070	−0.0133870
$477$	0	0
$478$	−28.3550	−1.29693
$479$	25.9098	1.18385	0.591923	−	0.805994i	$-0.298369\pi$
$479$	25.9098	1.18385	0.591923	+	0.805994i	$0.298369\pi$
$480$	0	0
$481$	−14.6137	−0.666327
$482$	21.9596	1.00023
$483$	0	0
$484$	−0.115460	−0.00524817
$485$	−2.95633	−0.134240
$486$	0	0
$487$	−3.09538	−0.140265	−0.0701326	−	0.997538i	$-0.522342\pi$
$487$	−3.09538	−0.140265	−0.0701326	+	0.997538i	$0.522342\pi$
$488$	16.2273	0.734577
$489$	0	0
$490$	1.48783	0.0672134
$491$	−17.4687	−0.788352	−0.394176	−	0.919035i	$-0.628970\pi$
$491$	−17.4687	−0.788352	−0.394176	+	0.919035i	$0.628970\pi$
$492$	0	0
$493$	−4.68718	−0.211100
$494$	−6.65547	−0.299444
$495$	0	0
$496$	−17.8902	−0.803293
$497$	0.631280	0.0283168
$498$	0	0
$499$	−6.47276	−0.289760	−0.144880	−	0.989449i	$-0.546280\pi$
$499$	−6.47276	−0.289760	−0.144880	+	0.989449i	$0.546280\pi$
$500$	0.115460	0.00516351
$501$	0	0
$502$	15.5644	0.694675
$503$	15.5931	0.695262	0.347631	−	0.937631i	$-0.386986\pi$
$503$	15.5931	0.695262	0.347631	+	0.937631i	$0.386986\pi$
$504$	0	0
$505$	−1.55010	−0.0689786
$506$	−0.517582	−0.0230093
$507$	0	0
$508$	−1.70719	−0.0757444
$509$	−6.14404	−0.272330	−0.136165	−	0.990686i	$-0.543478\pi$
$509$	−6.14404	−0.272330	−0.136165	+	0.990686i	$0.543478\pi$
$510$	0	0
$511$	−15.2673	−0.675388
$512$	−24.2638	−1.07232
$513$	0	0
$514$	−13.4536	−0.593414
$515$	0.339313	0.0149519
$516$	0	0
$517$	9.67305	0.425420
$518$	10.0649	0.442225
$519$	0	0
$520$	−14.0794	−0.617422
$521$	−41.4684	−1.81676	−0.908382	−	0.418141i	$-0.862682\pi$
$521$	−41.4684	−1.81676	−0.908382	+	0.418141i	$0.862682\pi$
$522$	0	0
$523$	2.76287	0.120812	0.0604059	−	0.998174i	$-0.480760\pi$
$523$	2.76287	0.120812	0.0604059	+	0.998174i	$0.480760\pi$
$524$	−0.902970	−0.0394464
$525$	0	0
$526$	29.2264	1.27433
$527$	4.95398	0.215799
$528$	0	0
$529$	−22.8578	−0.993819
$530$	−17.4982	−0.760075
$531$	0	0
$532$	−0.280835	−0.0121758
$533$	3.83222	0.165992
$534$	0	0
$535$	−9.05124	−0.391319
$536$	3.50603	0.151438
$537$	0	0
$538$	25.0929	1.08183
$539$	−1.08380	−0.0466828
$540$	0	0
$541$	29.6566	1.27504	0.637519	−	0.770435i	$-0.279961\pi$
$541$	29.6566	1.27504	0.637519	+	0.770435i	$0.279961\pi$
$542$	27.1636	1.16678
$543$	0	0
$544$	0.678402	0.0290862
$545$	−14.7515	−0.631886
$546$	0	0
$547$	−21.7318	−0.929184	−0.464592	−	0.885525i	$-0.653799\pi$
$547$	−21.7318	−0.929184	−0.464592	+	0.885525i	$0.653799\pi$
$548$	−0.889761	−0.0380087
$549$	0	0
$550$	1.37279	0.0585358
$551$	−4.50688	−0.192000
$552$	0	0
$553$	−22.3026	−0.948403
$554$	33.7236	1.43278
$555$	0	0
$556$	−1.66207	−0.0704876
$557$	−8.08244	−0.342464	−0.171232	−	0.985231i	$-0.554775\pi$
$557$	−8.08244	−0.342464	−0.171232	+	0.985231i	$0.554775\pi$
$558$	0	0
$559$	36.2049	1.53130
$560$	9.13520	0.386033
$561$	0	0
$562$	−19.3422	−0.815902
$563$	25.7685	1.08601	0.543006	−	0.839729i	$-0.317286\pi$
$563$	25.7685	1.08601	0.543006	+	0.839729i	$0.317286\pi$
$564$	0	0
$565$	−4.71030	−0.198164
$566$	−25.8588	−1.08693
$567$	0	0
$568$	−0.753717	−0.0316253
$569$	19.2047	0.805102	0.402551	−	0.915398i	$-0.368124\pi$
$569$	19.2047	0.805102	0.402551	+	0.915398i	$0.368124\pi$
$570$	0	0
$571$	−28.0284	−1.17295	−0.586477	−	0.809966i	$-0.699486\pi$
$571$	−28.0284	−1.17295	−0.586477	+	0.809966i	$0.699486\pi$
$572$	0.559766	0.0234050
$573$	0	0
$574$	−2.63936	−0.110165
$575$	−0.377030	−0.0157233
$576$	0	0
$577$	−34.8547	−1.45102	−0.725509	−	0.688212i	$-0.758396\pi$
$577$	−34.8547	−1.45102	−0.725509	+	0.688212i	$0.758396\pi$
$578$	−21.8525	−0.908946
$579$	0	0
$580$	−0.520363	−0.0216069
$581$	34.6436	1.43726
$582$	0	0
$583$	12.7465	0.527907
$584$	18.2285	0.754299
$585$	0	0
$586$	−2.85910	−0.118109
$587$	−7.98152	−0.329433	−0.164716	−	0.986341i	$-0.552671\pi$
$587$	−7.98152	−0.329433	−0.164716	+	0.986341i	$0.552671\pi$
$588$	0	0
$589$	4.76341	0.196273
$590$	1.81360	0.0746647
$591$	0	0
$592$	−11.3209	−0.465286
$593$	−11.6876	−0.479951	−0.239976	−	0.970779i	$-0.577139\pi$
$593$	−11.6876	−0.479951	−0.239976	+	0.970779i	$0.577139\pi$
$594$	0	0
$595$	−2.52963	−0.103705
$596$	−1.59796	−0.0654548
$597$	0	0
$598$	2.50931	0.102614
$599$	20.3132	0.829973	0.414986	−	0.909828i	$-0.363786\pi$
$599$	20.3132	0.829973	0.414986	+	0.909828i	$0.363786\pi$
$600$	0	0
$601$	32.4805	1.32491	0.662453	−	0.749104i	$-0.269516\pi$
$601$	32.4805	1.32491	0.662453	+	0.749104i	$0.269516\pi$
$602$	−24.9353	−1.01629
$603$	0	0
$604$	0.935548	0.0380669
$605$	−1.00000	−0.0406558
$606$	0	0
$607$	−45.4701	−1.84557	−0.922787	−	0.385311i	$-0.874094\pi$
$607$	−45.4701	−1.84557	−0.922787	+	0.385311i	$0.874094\pi$
$608$	0.652306	0.0264545
$609$	0	0
$610$	7.67083	0.310583
$611$	−46.8964	−1.89722
$612$	0	0
$613$	4.06460	0.164168	0.0820839	−	0.996625i	$-0.473842\pi$
$613$	4.06460	0.164168	0.0820839	+	0.996625i	$0.473842\pi$
$614$	27.2418	1.09939
$615$	0	0
$616$	−7.06364	−0.284602
$617$	27.7980	1.11911	0.559554	−	0.828794i	$-0.310973\pi$
$617$	27.7980	1.11911	0.559554	+	0.828794i	$0.310973\pi$
$618$	0	0
$619$	31.8288	1.27931	0.639654	−	0.768663i	$-0.279077\pi$
$619$	31.8288	1.27931	0.639654	+	0.768663i	$0.279077\pi$
$620$	0.549982	0.0220878
$621$	0	0
$622$	−5.10420	−0.204660
$623$	3.01566	0.120820
$624$	0	0
$625$	1.00000	0.0400000
$626$	37.1840	1.48617
$627$	0	0
$628$	0.823665	0.0328678
$629$	3.13487	0.124995
$630$	0	0
$631$	32.8526	1.30784	0.653920	−	0.756564i	$-0.273123\pi$
$631$	32.8526	1.30784	0.653920	+	0.756564i	$0.273123\pi$
$632$	26.6282	1.05921
$633$	0	0
$634$	−13.0476	−0.518185
$635$	−14.7860	−0.586766
$636$	0	0
$637$	5.25445	0.208189
$638$	−6.18698	−0.244945
$639$	0	0
$640$	−10.2364	−0.404628
$641$	−31.2727	−1.23520	−0.617599	−	0.786493i	$-0.711894\pi$
$641$	−31.2727	−1.23520	−0.617599	+	0.786493i	$0.711894\pi$
$642$	0	0
$643$	−13.7579	−0.542557	−0.271279	−	0.962501i	$-0.587446\pi$
$643$	−13.7579	−0.542557	−0.271279	+	0.962501i	$0.587446\pi$
$644$	0.105883	0.00417239
$645$	0	0
$646$	1.42770	0.0561723
$647$	−20.3525	−0.800140	−0.400070	−	0.916485i	$-0.631014\pi$
$647$	−20.3525	−0.800140	−0.400070	+	0.916485i	$0.631014\pi$
$648$	0	0
$649$	−1.32111	−0.0518580
$650$	−6.65547	−0.261049
$651$	0	0
$652$	2.75608	0.107936
$653$	4.04869	0.158437	0.0792187	−	0.996857i	$-0.474757\pi$
$653$	4.04869	0.158437	0.0792187	+	0.996857i	$0.474757\pi$
$654$	0	0
$655$	−7.82065	−0.305578
$656$	2.96873	0.115910
$657$	0	0
$658$	32.2989	1.25914
$659$	13.0892	0.509882	0.254941	−	0.966957i	$-0.417944\pi$
$659$	13.0892	0.509882	0.254941	+	0.966957i	$0.417944\pi$
$660$	0	0
$661$	9.55227	0.371540	0.185770	−	0.982593i	$-0.440522\pi$
$661$	9.55227	0.371540	0.185770	+	0.982593i	$0.440522\pi$
$662$	9.99208	0.388353
$663$	0	0
$664$	−41.3627	−1.60518
$665$	−2.43232	−0.0943214
$666$	0	0
$667$	1.69923	0.0657945
$668$	−0.981386	−0.0379710
$669$	0	0
$670$	1.65734	0.0640286
$671$	−5.58779	−0.215714
$672$	0	0
$673$	31.4730	1.21320	0.606598	−	0.795009i	$-0.292534\pi$
$673$	31.4730	1.21320	0.606598	+	0.795009i	$0.292534\pi$
$674$	40.3310	1.55349
$675$	0	0
$676$	−1.21285	−0.0466482
$677$	35.4776	1.36352	0.681758	−	0.731578i	$-0.261216\pi$
$677$	35.4776	1.36352	0.681758	+	0.731578i	$0.261216\pi$
$678$	0	0
$679$	7.19075	0.275956
$680$	3.02025	0.115821
$681$	0	0
$682$	6.53915	0.250397
$683$	40.6216	1.55434	0.777171	−	0.629289i	$-0.216654\pi$
$683$	40.6216	1.55434	0.777171	+	0.629289i	$0.216654\pi$
$684$	0	0
$685$	−7.70625	−0.294441
$686$	−26.9923	−1.03057
$687$	0	0
$688$	28.0471	1.06929
$689$	−61.7971	−2.35428
$690$	0	0
$691$	−12.5025	−0.475616	−0.237808	−	0.971312i	$-0.576429\pi$
$691$	−12.5025	−0.475616	−0.237808	+	0.971312i	$0.576429\pi$
$692$	−0.434974	−0.0165352
$693$	0	0
$694$	−38.5791	−1.46444
$695$	−14.3953	−0.546044
$696$	0	0
$697$	−0.822073	−0.0311382
$698$	28.8214	1.09091
$699$	0	0
$700$	−0.280835	−0.0106146
$701$	22.1578	0.836887	0.418443	−	0.908243i	$-0.362576\pi$
$701$	22.1578	0.836887	0.418443	+	0.908243i	$0.362576\pi$
$702$	0	0
$703$	3.01428	0.113686
$704$	8.40698	0.316850
$705$	0	0
$706$	−20.8491	−0.784665
$707$	3.77035	0.141798
$708$	0	0
$709$	−32.1717	−1.20824	−0.604118	−	0.796895i	$-0.706474\pi$
$709$	−32.1717	−1.20824	−0.604118	+	0.796895i	$0.706474\pi$
$710$	−0.356290	−0.0133713
$711$	0	0
$712$	−3.60055	−0.134936
$713$	−1.79595	−0.0672589
$714$	0	0
$715$	4.84815	0.181311
$716$	−1.38922	−0.0519178
$717$	0	0
$718$	−15.7908	−0.589308
$719$	−4.63511	−0.172860	−0.0864302	−	0.996258i	$-0.527546\pi$
$719$	−4.63511	−0.172860	−0.0864302	+	0.996258i	$0.527546\pi$
$720$	0	0
$721$	−0.825320	−0.0307365
$722$	1.37279	0.0510898
$723$	0	0
$724$	1.04134	0.0387009
$725$	−4.50688	−0.167381
$726$	0	0
$727$	−29.6755	−1.10060	−0.550302	−	0.834966i	$-0.685487\pi$
$727$	−29.6755	−1.10060	−0.550302	+	0.834966i	$0.685487\pi$
$728$	34.2456	1.26923
$729$	0	0
$730$	8.61678	0.318921
$731$	−7.76652	−0.287255
$732$	0	0
$733$	36.0642	1.33206	0.666031	−	0.745924i	$-0.267992\pi$
$733$	36.0642	1.33206	0.666031	+	0.745924i	$0.267992\pi$
$734$	−2.09150	−0.0771987
$735$	0	0
$736$	−0.245939	−0.00906544
$737$	−1.20728	−0.0444708
$738$	0	0
$739$	13.1048	0.482066	0.241033	−	0.970517i	$-0.422514\pi$
$739$	13.1048	0.482066	0.241033	+	0.970517i	$0.422514\pi$
$740$	0.348028	0.0127938
$741$	0	0
$742$	42.5614	1.56248
$743$	−49.4215	−1.81310	−0.906550	−	0.422098i	$-0.861294\pi$
$743$	−49.4215	−1.81310	−0.906550	+	0.422098i	$0.861294\pi$
$744$	0	0
$745$	−13.8399	−0.507056
$746$	22.7396	0.832557
$747$	0	0
$748$	−0.120079	−0.00439051
$749$	22.0155	0.804431
$750$	0	0
$751$	5.10832	0.186405	0.0932026	−	0.995647i	$-0.470290\pi$
$751$	5.10832	0.186405	0.0932026	+	0.995647i	$0.470290\pi$
$752$	−36.3295	−1.32480
$753$	0	0
$754$	29.9954	1.09237
$755$	8.10281	0.294892
$756$	0	0
$757$	−38.3696	−1.39457	−0.697283	−	0.716796i	$-0.745608\pi$
$757$	−38.3696	−1.39457	−0.697283	+	0.716796i	$0.745608\pi$
$758$	−0.0261656	−0.000950378 0
$759$	0	0
$760$	2.90407	0.105342
$761$	34.9845	1.26819	0.634093	−	0.773257i	$-0.281374\pi$
$761$	34.9845	1.26819	0.634093	+	0.773257i	$0.281374\pi$
$762$	0	0
$763$	35.8805	1.29896
$764$	−1.76189	−0.0637428
$765$	0	0
$766$	−24.9964	−0.903157
$767$	6.40493	0.231269
$768$	0	0
$769$	4.72143	0.170259	0.0851296	−	0.996370i	$-0.472870\pi$
$769$	4.72143	0.170259	0.0851296	+	0.996370i	$0.472870\pi$
$770$	−3.33906	−0.120331
$771$	0	0
$772$	−0.366292	−0.0131831
$773$	−47.5946	−1.71186	−0.855930	−	0.517092i	$-0.827014\pi$
$773$	−47.5946	−1.71186	−0.855930	+	0.517092i	$0.827014\pi$
$774$	0	0
$775$	4.76341	0.171107
$776$	−8.58539	−0.308198
$777$	0	0
$778$	43.1097	1.54556
$779$	−0.790451	−0.0283208
$780$	0	0
$781$	0.259538	0.00928700
$782$	−0.538288	−0.0192491
$783$	0	0
$784$	4.07050	0.145375
$785$	7.13378	0.254616
$786$	0	0
$787$	1.02158	0.0364154	0.0182077	−	0.999834i	$-0.494204\pi$
$787$	1.02158	0.0364154	0.0182077	+	0.999834i	$0.494204\pi$
$788$	−0.703608	−0.0250650
$789$	0	0
$790$	12.5874	0.447841
$791$	11.4570	0.407363
$792$	0	0
$793$	27.0904	0.962010
$794$	−35.5028	−1.25995
$795$	0	0
$796$	1.82597	0.0647199
$797$	37.8469	1.34061	0.670304	−	0.742087i	$-0.266164\pi$
$797$	37.8469	1.34061	0.670304	+	0.742087i	$0.266164\pi$
$798$	0	0
$799$	10.0600	0.355898
$800$	0.652306	0.0230625
$801$	0	0
$802$	21.2975	0.752042
$803$	−6.27686	−0.221506
$804$	0	0
$805$	0.917060	0.0323221
$806$	−31.7028	−1.11668
$807$	0	0
$808$	−4.50161	−0.158366
$809$	−25.6686	−0.902461	−0.451231	−	0.892407i	$-0.649015\pi$
$809$	−25.6686	−0.902461	−0.451231	+	0.892407i	$0.649015\pi$
$810$	0	0
$811$	−6.16852	−0.216606	−0.108303	−	0.994118i	$-0.534542\pi$
$811$	−6.16852	−0.216606	−0.108303	+	0.994118i	$0.534542\pi$
$812$	1.26569	0.0444170
$813$	0	0
$814$	4.13797	0.145036
$815$	23.8705	0.836147
$816$	0	0
$817$	−7.46777	−0.261264
$818$	28.4785	0.995729
$819$	0	0
$820$	−0.0912652	−0.00318712
$821$	30.9910	1.08159	0.540796	−	0.841154i	$-0.318123\pi$
$821$	30.9910	1.08159	0.540796	+	0.841154i	$0.318123\pi$
$822$	0	0
$823$	−40.7706	−1.42117	−0.710587	−	0.703609i	$-0.751571\pi$
$823$	−40.7706	−1.42117	−0.710587	+	0.703609i	$0.751571\pi$
$824$	0.985391	0.0343277
$825$	0	0
$826$	−4.41126	−0.153487
$827$	−46.2940	−1.60980	−0.804900	−	0.593410i	$-0.797781\pi$
$827$	−46.2940	−1.60980	−0.804900	+	0.593410i	$0.797781\pi$
$828$	0	0
$829$	−14.8471	−0.515660	−0.257830	−	0.966190i	$-0.583007\pi$
$829$	−14.8471	−0.515660	−0.257830	+	0.966190i	$0.583007\pi$
$830$	−19.5526	−0.678680
$831$	0	0
$832$	−40.7583	−1.41304
$833$	−1.12716	−0.0390539
$834$	0	0
$835$	−8.49981	−0.294148
$836$	−0.115460	−0.00399326
$837$	0	0
$838$	−9.54024	−0.329562
$839$	−36.8854	−1.27343	−0.636713	−	0.771101i	$-0.719706\pi$
$839$	−36.8854	−1.27343	−0.636713	+	0.771101i	$0.719706\pi$
$840$	0	0
$841$	−8.68803	−0.299587
$842$	23.4040	0.806556
$843$	0	0
$844$	−1.20507	−0.0414802
$845$	−10.5046	−0.361368
$846$	0	0
$847$	2.43232	0.0835756
$848$	−47.8728	−1.64396
$849$	0	0
$850$	1.42770	0.0489698
$851$	−1.13648	−0.0389579
$852$	0	0
$853$	9.84716	0.337160	0.168580	−	0.985688i	$-0.446082\pi$
$853$	9.84716	0.337160	0.168580	+	0.985688i	$0.446082\pi$
$854$	−18.6579	−0.638462
$855$	0	0
$856$	−26.2855	−0.898419
$857$	25.6874	0.877464	0.438732	−	0.898618i	$-0.355428\pi$
$857$	25.6874	0.877464	0.438732	+	0.898618i	$0.355428\pi$
$858$	0	0
$859$	48.4118	1.65179	0.825894	−	0.563825i	$-0.190671\pi$
$859$	48.4118	1.65179	0.825894	+	0.563825i	$0.190671\pi$
$860$	−0.862227	−0.0294017
$861$	0	0
$862$	−14.5754	−0.496439
$863$	−15.3028	−0.520913	−0.260457	−	0.965486i	$-0.583873\pi$
$863$	−15.3028	−0.520913	−0.260457	+	0.965486i	$0.583873\pi$
$864$	0	0
$865$	−3.76732	−0.128093
$866$	−7.16528	−0.243486
$867$	0	0
$868$	−1.33773	−0.0454057
$869$	−9.16926	−0.311046
$870$	0	0
$871$	5.85308	0.198324
$872$	−42.8395	−1.45073
$873$	0	0
$874$	−0.517582	−0.0175075
$875$	−2.43232	−0.0822275
$876$	0	0
$877$	−34.7213	−1.17245	−0.586227	−	0.810147i	$-0.699387\pi$
$877$	−34.7213	−1.17245	−0.586227	+	0.810147i	$0.699387\pi$
$878$	−35.4310	−1.19574
$879$	0	0
$880$	3.75575	0.126606
$881$	−33.7901	−1.13842	−0.569208	−	0.822194i	$-0.692750\pi$
$881$	−33.7901	−1.13842	−0.569208	+	0.822194i	$0.692750\pi$
$882$	0	0
$883$	54.4286	1.83167	0.915835	−	0.401556i	$-0.131530\pi$
$883$	54.4286	1.83167	0.915835	+	0.401556i	$0.131530\pi$
$884$	0.582160	0.0195801
$885$	0	0
$886$	41.7213	1.40166
$887$	−14.5748	−0.489373	−0.244686	−	0.969602i	$-0.578685\pi$
$887$	−14.5748	−0.489373	−0.244686	+	0.969602i	$0.578685\pi$
$888$	0	0
$889$	35.9644	1.20621
$890$	−1.70202	−0.0570517
$891$	0	0
$892$	1.75685	0.0588237
$893$	9.67305	0.323696
$894$	0	0
$895$	−12.0321	−0.402189
$896$	24.8981	0.831789
$897$	0	0
$898$	−14.9049	−0.497384
$899$	−21.4681	−0.716003
$900$	0	0
$901$	13.2565	0.441636
$902$	−1.08512	−0.0361305
$903$	0	0
$904$	−13.6791	−0.454959
$905$	9.01903	0.299803
$906$	0	0
$907$	−36.7115	−1.21898	−0.609492	−	0.792792i	$-0.708627\pi$
$907$	−36.7115	−1.21898	−0.609492	+	0.792792i	$0.708627\pi$
$908$	−1.78004	−0.0590727
$909$	0	0
$910$	16.1883	0.536635
$911$	−18.5321	−0.613997	−0.306999	−	0.951710i	$-0.599325\pi$
$911$	−18.5321	−0.613997	−0.306999	+	0.951710i	$0.599325\pi$
$912$	0	0
$913$	14.2430	0.471374
$914$	25.2385	0.834815
$915$	0	0
$916$	−2.10747	−0.0696328
$917$	19.0224	0.628173
$918$	0	0
$919$	−30.1650	−0.995052	−0.497526	−	0.867449i	$-0.665758\pi$
$919$	−30.1650	−0.995052	−0.497526	+	0.867449i	$0.665758\pi$
$920$	−1.09492	−0.0360986
$921$	0	0
$922$	38.8035	1.27792
$923$	−1.25828	−0.0414167
$924$	0	0
$925$	3.01428	0.0991090
$926$	45.8875	1.50796
$927$	0	0
$928$	−2.93987	−0.0965058
$929$	53.1253	1.74298	0.871492	−	0.490409i	$-0.163153\pi$
$929$	53.1253	1.74298	0.871492	+	0.490409i	$0.163153\pi$
$930$	0	0
$931$	−1.08380	−0.0355203
$932$	2.20575	0.0722519
$933$	0	0
$934$	1.66515	0.0544854
$935$	−1.04001	−0.0340118
$936$	0	0
$937$	46.9097	1.53247	0.766237	−	0.642558i	$-0.222127\pi$
$937$	46.9097	1.53247	0.766237	+	0.642558i	$0.222127\pi$
$938$	−4.03118	−0.131623
$939$	0	0
$940$	1.11685	0.0364275
$941$	−36.0038	−1.17369	−0.586845	−	0.809699i	$-0.699630\pi$
$941$	−36.0038	−1.17369	−0.586845	+	0.809699i	$0.699630\pi$
$942$	0	0
$943$	0.298024	0.00970499
$944$	4.96175	0.161491
$945$	0	0
$946$	−10.2517	−0.333310
$947$	21.3336	0.693249	0.346625	−	0.938004i	$-0.387328\pi$
$947$	21.3336	0.693249	0.346625	+	0.938004i	$0.387328\pi$
$948$	0	0
$949$	30.4312	0.987837
$950$	1.37279	0.0445390
$951$	0	0
$952$	−7.34622	−0.238093
$953$	−0.777926	−0.0251995	−0.0125997	−	0.999921i	$-0.504011\pi$
$953$	−0.777926	−0.0251995	−0.0125997	+	0.999921i	$0.504011\pi$
$954$	0	0
$955$	−15.2598	−0.493794
$956$	2.38483	0.0771310
$957$	0	0
$958$	35.5685	1.14917
$959$	18.7441	0.605278
$960$	0	0
$961$	−8.30988	−0.268061
$962$	−20.0615	−0.646808
$963$	0	0
$964$	−1.84693	−0.0594857
$965$	−3.17246	−0.102125
$966$	0	0
$967$	41.5502	1.33616	0.668082	−	0.744088i	$-0.267116\pi$
$967$	41.5502	1.33616	0.668082	+	0.744088i	$0.267116\pi$
$968$	−2.90407	−0.0933404
$969$	0	0
$970$	−4.05841	−0.130308
$971$	−41.5722	−1.33412	−0.667058	−	0.745006i	$-0.732447\pi$
$971$	−41.5722	−1.33412	−0.667058	+	0.745006i	$0.732447\pi$
$972$	0	0
$973$	35.0139	1.12250
$974$	−4.24930	−0.136156
$975$	0	0
$976$	20.9863	0.671756
$977$	4.43754	0.141969	0.0709847	−	0.997477i	$-0.477386\pi$
$977$	4.43754	0.141969	0.0709847	+	0.997477i	$0.477386\pi$
$978$	0	0
$979$	1.23983	0.0396250
$980$	−0.125136	−0.00399732
$981$	0	0
$982$	−23.9808	−0.765258
$983$	−4.98576	−0.159021	−0.0795106	−	0.996834i	$-0.525336\pi$
$983$	−4.98576	−0.159021	−0.0795106	+	0.996834i	$0.525336\pi$
$984$	0	0
$985$	−6.09397	−0.194170
$986$	−6.43449	−0.204916
$987$	0	0
$988$	0.559766	0.0178085
$989$	2.81558	0.0895301
$990$	0	0
$991$	−39.9837	−1.27012	−0.635062	−	0.772461i	$-0.719025\pi$
$991$	−39.9837	−1.27012	−0.635062	+	0.772461i	$0.719025\pi$
$992$	3.10720	0.0986539
$993$	0	0
$994$	0.866612	0.0274873
$995$	15.8148	0.501363
$996$	0	0
$997$	−31.3940	−0.994258	−0.497129	−	0.867677i	$-0.665612\pi$
$997$	−31.3940	−0.994258	−0.497129	+	0.867677i	$0.665612\pi$
$998$	−8.88571	−0.281272
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9405.2.a.v.1.3		5
3.2	odd	2		1045.2.a.d.1.3	✓	5
15.14	odd	2		5225.2.a.j.1.3		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1045.2.a.d.1.3	✓	5	3.2	odd	2
5225.2.a.j.1.3		5	15.14	odd	2
9405.2.a.v.1.3		5	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 9405 = 3^{2} \cdot 5 \cdot 11 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9405.a (trivial)

\(f(q)\)	\(=\)	\(q+1.37279 q^{2} -0.115460 q^{4} -1.00000 q^{5} +2.43232 q^{7} -2.90407 q^{8} -1.37279 q^{10} +1.00000 q^{11} -4.84815 q^{13} +3.33906 q^{14} -3.75575 q^{16} +1.04001 q^{17} +1.00000 q^{19} +0.115460 q^{20} +1.37279 q^{22} -0.377030 q^{23} +1.00000 q^{25} -6.65547 q^{26} -0.280835 q^{28} -4.50688 q^{29} +4.76341 q^{31} +0.652306 q^{32} +1.42770 q^{34} -2.43232 q^{35} +3.01428 q^{37} +1.37279 q^{38} +2.90407 q^{40} -0.790451 q^{41} -7.46777 q^{43} -0.115460 q^{44} -0.517582 q^{46} +9.67305 q^{47} -1.08380 q^{49} +1.37279 q^{50} +0.559766 q^{52} +12.7465 q^{53} -1.00000 q^{55} -7.06364 q^{56} -6.18698 q^{58} -1.32111 q^{59} -5.58779 q^{61} +6.53915 q^{62} +8.40698 q^{64} +4.84815 q^{65} -1.20728 q^{67} -0.120079 q^{68} -3.33906 q^{70} +0.259538 q^{71} -6.27686 q^{73} +4.13797 q^{74} -0.115460 q^{76} +2.43232 q^{77} -9.16926 q^{79} +3.75575 q^{80} -1.08512 q^{82} +14.2430 q^{83} -1.04001 q^{85} -10.2517 q^{86} -2.90407 q^{88} +1.23983 q^{89} -11.7923 q^{91} +0.0435318 q^{92} +13.2790 q^{94} -1.00000 q^{95} +2.95633 q^{97} -1.48783 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 3 q^{2} + 5 q^{4} - 5 q^{5} - 11 q^{7} - 3 q^{8} - 3 q^{10} + 5 q^{11} + q^{13} - 3 q^{16} + 3 q^{17} + 5 q^{19} - 5 q^{20} + 3 q^{22} + 8 q^{23} + 5 q^{25} + 16 q^{26} - 22 q^{28} - 11 q^{29} - 5 q^{31}+ \cdots - 10 q^{98}+O(q^{100}) \) 5 * q + 3 * q^2 + 5 * q^4 - 5 * q^5 - 11 * q^7 - 3 * q^8 - 3 * q^10 + 5 * q^11 + q^13 - 3 * q^16 + 3 * q^17 + 5 * q^19 - 5 * q^20 + 3 * q^22 + 8 * q^23 + 5 * q^25 + 16 * q^26 - 22 * q^28 - 11 * q^29 - 5 * q^31 + 2 * q^32 + 4 * q^34 + 11 * q^35 - 9 * q^37 + 3 * q^38 + 3 * q^40 - 15 * q^41 - 13 * q^43 + 5 * q^44 + 18 * q^46 + 20 * q^47 + 20 * q^49 + 3 * q^50 + q^52 + 5 * q^53 - 5 * q^55 - 33 * q^58 + 17 * q^59 + 3 * q^61 - 14 * q^62 - 17 * q^64 - q^65 - 28 * q^67 + 25 * q^68 + 6 * q^71 - 16 * q^73 + 21 * q^74 + 5 * q^76 - 11 * q^77 + 3 * q^79 + 3 * q^80 + 2 * q^82 + 33 * q^83 - 3 * q^85 - 10 * q^86 - 3 * q^88 + 16 * q^89 - 22 * q^91 + 19 * q^92 - 10 * q^94 - 5 * q^95 - 14 * q^97 - 10 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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